Fig. 1

Two pairwise alignments and a bi-alignment of peptide sequences and their predicted secondary structures (helix red, turn blue, \(\beta\)-sheet green, coil orange). Structure are predicted according to the Chou Fasman method [13] with CFSSP [14]. To facilitate quick visual assessment of sequence alignment quality, sequence mismatches are shown in bold black, sequence indels in non-bold black, and mismatches in dark red. The upper alignment optimizes sequence similarity, and shows the structure out of sync: the helix is moved to left, the last \(\beta\)-sheet is shifted to the right by 1 position. The second alignment maximizes structural similarity and thus shows little sequence similarity. The evolution of the two peptides is explained much better by a bi-alignment (third panel), which supports shift events (marked by rectangles) that can shift either sequence against its structure to the left (\(\texttt {<}\)) or to the right (\(\texttt {>}\)). The resulting regions of shift are indicated by in general k blue and red lines corresponding to shifts by k positions to the left or to the right. While the shift events shown in this example delete and insert structure of A with respect to both sequences and the structure of B, shift alignments also support as well analogous shifts of sequences and the second structure (which would be shown in the bottom row). In our representation, shift events are the only visible difference between the bi-alignment \(\mathbb {A}\) in the third panel and the two alignments. Nevertheless, the representation can be mapped to our formalization of bi-alignments as alignments of two constituent alignments \(\mathbb {U}\) and \(\mathbb {V}\): \(\mathbb {U}\) is obtained from the 2nd and 3rd bi-alignment row by removing the two all-gap columns (i.e. the first and the 3rd-to-last column). The secondary structure alignment \(\mathbb {V}\) coincides with the 1st and 4th row since there is no column that contains only gaps in these to rows